MinT - Best Known (182−122, 182, s)-Nets in Base 3

Best Known (182−122, 182, s)-Nets in Base 3

(182−122, 182, 48)-Net over F₃ — Constructive and digital

Digital (60, 182, 48)-net over F₃, using

t-expansion [i] based on digital (45, 182, 48)-net over F₃, using
- net from sequence [i] based on digital (45, 47)-sequence over F₃, using
  - Niederreiterâ€“Xing sequence constructionÂ II/III [i] based on function field F/F₃ with g(F) = 45 and N(F) ≥ 48, using
    - an explicitly constructive algebraic function field [i]

(182−122, 182, 64)-Net over F₃ — Digital

Digital (60, 182, 64)-net over F₃, using

t-expansion [i] based on digital (49, 182, 64)-net over F₃, using
- net from sequence [i] based on digital (49, 63)-sequence over F₃, using
  - Niederreiterâ€“Xing sequence constructionÂ II/III [i] based on function field F/F₃ with g(F) = 49 and N(F) ≥ 64, using
    - a curve from the manYPoints database [i]

(182−122, 182, 191)-Net over F₃ — Upper bound on s (digital)

There is no digital (60, 182, 192)-net over F₃, because

2 times m-reduction [i] would yield digital (60, 180, 192)-net over F₃, but
- extracting embedded orthogonal array [i] would yield linear OA(3¹⁸⁰, 192, F₃, 120) (dual of [192, 12, 121]-code), but
  - residual code [i] would yield linear OA(3⁶⁰, 71, F₃, 40) (dual of [71, 11, 41]-code), but
    - â€œGurâ€ bound on codes from Brouwerâ€™s database [i]

(182−122, 182, 256)-Net in Base 3 — Upper bound on s

There is no (60, 182, 257)-net in base 3, because

the generalized Rao bound for nets shows that 3^mÂ â‰¥ 781 175611 443379 070895 810035 654264 084185 027808 609504 416115 703310 742324 995556 156842 693395 > 3¹⁸² [i]